Express $z_1=3[\cos(60^{\circ})+i\sin(60^{\circ})]$ in rectangular form. Express your answer in exact terms. $z_1=$
Solution: The Strategy A complex number in rectangular form, $z={a}+{b}i$, can be written in polar form as $z={r}[\cos{\theta}+i\sin{\theta}]$, where ${r}$ is the absolute value, or modulus, and ${\theta}$ is the angle, or argument. Therefore, ${r}$ and ${\theta}$ can be found using the following formulas: ${r}=\sqrt{{a}^2+{b}^2}$ $\tan{\theta}=\dfrac{{b}}{{a}}$ [How did we get these equations?] Similarly, a complex number in polar form, $z={r}[\cos{\theta}+i\sin{\theta}]$, can be written in rectangular form as $z={a}+{b}i$, using the following formulas: ${a}={r}\cos{\theta}$ ${b}={r}\sin{\theta}$ [How did we get these equations?] Finding $a$ For $z_1={3}[\cos{60^\circ}+i\sin{60^\circ}]$ : ${r}={3}$ ${\theta}={60^\circ}$ Therefore, we can find ${a}$ as follows. $\begin{aligned}{a}&={r}\cos{\theta} \\\\&={3}\cos{60^\circ} \\\\&={\dfrac{3}{2}}\end{aligned}$ Finding $b$ $\begin{aligned}{b}&={r}\sin{\theta} \\\\&={3}\sin{60^\circ} \\\\&={\dfrac{3\sqrt{3}}{2}}\end{aligned}$ Summary $z_1={\dfrac{3}{2}}+{\dfrac{3\sqrt{3}}{2}}i$